Problem 60 - Algebra and Trigonometry

Question

\(59-64\) Find the domain of the function. $$ f(x)=\log _{5}(8-2 x) $$

Step-by-Step Solution

Answer

The domain of \( f(x) = \log_{5}(8-2x) \) is \( x < 4 \).

Step 1: Understanding the Domain of a Logarithm

For a function involving a logarithm like \( f(x) = \log_{5}(8-2x) \), the expression inside the logarithm, \( 8-2x \), must be positive. This is because the logarithm of a non-positive number is undefined.

Step 2: Setting Up the Inequality

To find the domain, set the expression inside the logarithm greater than zero: \( 8 - 2x > 0 \). This inequality will help determine the range of \( x \) values for which the function is defined.

Step 3: Solving the Inequality

Solve the inequality \( 8 - 2x > 0 \) to find \( x \): 1. Subtract 8 from both sides: \( -2x > -8 \). 2. Divide both sides by -2, remembering to reverse the inequality sign: \( x < 4 \).

Step 4: Defining the Domain

The solution to the inequality \( x < 4 \) indicates that \( x \) must be less than 4. Thus, the domain of the function is all real numbers less than 4.

Key Concepts

Logarithmic Functions

Logarithmic functions are powerful tools in mathematics, transforming multiplication and division into addition and subtraction. They are the inverse operations of exponential functions. In a logarithmic function like \( f(x) = \log_{5}(8-2x) \), it's essential to consider the base and the argument. Here, the base is 5. This means the function is looking for the power to which 5 must be raised to produce the expression inside the logarithm, \( 8-2x \).

For such functions to be valid, the argument, \( 8-2x \), must be positive because logarithms of zero or negative numbers don't exist in the real number system. This requirement leads us to form an inequality to ensure that the function stays within the realm of real numbers.

Inequalities

Inequalities are mathematical statements that relate expressions with greater than, less than, or equal symbols. To find the domain of a logarithm, we often set up an inequality, as seen in the exercise \( 8 - 2x > 0 \). This inequality ensures the argument of the logarithm is positive.

Solving inequalities involves similar steps to solving equations, but with special attention to direction changes. Specifically, when multiplying or dividing by a negative, flip the inequality sign. In our example, start by subtracting 8 from both sides to get \( -2x > -8 \). Then, divide by -2, flipping the sign to \( x < 4 \). This step reveals the range of permissible \( x \) values for the function.

Real Numbers

Real numbers include all the numbers on the number line, such as integers, fractions, and irrationals, excluding imaginary numbers. In the context of this exercise, real numbers form the potential domain of the function.

When determining the domain of \( f(x) = \log_{5}(8-2x) \), we've found \( x \) must be less than 4. This means the domain is all real numbers below 4, denoted as \( (-\infty, 4) \). It's crucial to understand that this domain excludes 4 itself, as substituting 4 results in 0 inside the logarithm, which is undefined for real numbers.

Visualizing the domain on a number line helps: all values to the left of 4 are included, capturing every point along the line except where \( x = 4 \).